A tauberian approach to RH

TL;DR

This paper introduces the concept of functions of good variation (FGV) inspired by Tauberian theory, proposing a novel approach to prove the Riemann Hypothesis (RH) through conjectures and experimental evidence.

Contribution

It defines FGV and demonstrates their potential to prove RH, offering new conjectural methods inspired by Tauberian theory and experimental mathematics.

Findings

Introduction of functions of good variation (FGV)

Experimental support for FGV approaching specific functions

Proposed conjectures linking FGV to RH proof

Abstract

The aim of this paper is twofold. Firstly we present our main discovery arising from experiments which is the tauberian concept of functions of good variation (FGV). Secondly we propose to use these FGV for proving RH is true via some conjectures. More precisely we give an implicit definition of FGV and we provide several smooth and nontrivial exemples from experiments. Then using a conjectured family of FGV approaching the function $x\mapsto x^{-1}\lfloor x\rfloor$ we derive RH is true. We make also a tauberian conjecture allowing us to prove RH is true for infinitely many $L$-functions and we discuss the linear independance conjecture. The method is inspired by the Ingham summation process and the experimental support is provided using pari-gp